On entire functions of fast growth

Abstract:

Let \[ (\ast )\quad f(z) = \sum \limits _{n = 0}^\infty {{a_n}{z^{{\lambda _n}}}} \] be a transcendental entire function. Set \[ M(r) = \max _{|z| = r} |f(z)|,\; m(r) = \max _{n \geq 0} \{ |{a_n}|{r^{{\lambda _n}}}\} \] and \[ N(r) = \max _{n \geq 0} \{ {\lambda _n}|m(r) = |{a_n}|{r^{{\lambda _n}}}\} .\] Sato introduced the notion of growth constants, referred in the present paper as ${S_q}$-order $\lambda$ and ${S_q}$-type $T$, which are generalizations of concepts of classical order and type by defining \[ (\ast \ast )\quad \lambda = \lim _{r \to \infty } \sup ({\log ^{[q]}}M(r)|\log r)\] and if $0 < \lambda < \infty$, then \[ (\ast \ast \ast )\quad T = \lim _{r \to \infty } \sup ({\log ^{[q - 1]}}M(r)|{r^\lambda })\] for $q = 2,3,4, \cdots$ where ${\log ^{[0]}}x = x$ and ${\log ^{[q]}}x = \log ({\log ^{[q - 1]}}x)$. Sato has also obtained the coefficient equivalents of $(\ast \ast )$ and $(\ast \ast \ast )$ for the entire function $(\ast )$ when ${\lambda _n} = n$. It is noted that Sato’s coefficient equivalents of $\lambda$, and $T$ also hold true for $(\ast )$ if $n$’s are replaced by ${\lambda _n}$’s in his coefficient equivalents. Analogous to $(\ast \ast )$ and $(\ast \ast \ast )$ lower ${S_q}$-order $v$ and lower ${S_q}$-type $t$ for entire function $f(z)$ are introduced here by defining \[ v = \lim _{r \to \infty } \inf ({\log ^{[q]}}M(r)|\log r)\] and if $0 < \lambda < \infty$ then \[ t = \lim _{r \to \infty } \inf ({\log ^{[q - 1]}}M(r)|{r^\lambda }),\quad q = 2,3,4, \cdots .\] For the case $q = 2$, these notions are due to Whittakar and Shah respectively. For the constant $v$, two complete coefficient characterizations have been found which generalize the earlier known results. For $t$ coefficient characterization only for those entire functions for which the consecutive principal indices are asymptotic is obtained. Determination of a complete coefficient characterization of $t$ remains an open problem. Further ${S_q}$-growth and lower ${S_q}$-growth numbers for entire function $f(z)$ we defined \[ \begin {array}{*{20}{c}} \delta \\ \mu \\ \end {array} = \lim _{r \to \infty } \begin {array}{*{20}{c}} {\sup } \\ {\inf } \\ \end {array} ({\log ^{[q - 1]}}N(r)|{r^\lambda }),\] for $q = 2,3,4, \cdots$ and $0 < \lambda < \infty$. Earlier results of Juneja giving the coefficients characterization of $\delta$ and $\mu$ are extended and generalized. A new decomposition theorem for entire functions of ${S_q}$-regular growth but not of perfectly ${S_q}$-regular growth has been found.